Gravity(gravitational waves)they say travels at the speed of light. But speed is distance/time. If gravity affects space-time or “time” then how can we be sure that gravity indeed travels with a quantity that gets affected by itself ?
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2$\begingroup$ Gravity doesn't affect space-time, mass affects space-time, and 'gravity' is the word we use to describe this interaction. $\endgroup$– WolfieSmithMay 4, 2022 at 12:59
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In the theory of general relativity, gravity is equivalent to inertia as the equation of motion is given by the free-fall/geodesic equation in a curved 4-dimensional spacetime. The gravitational potential is typically identified as being represented by the metric of the spacetime manifold. Rays of light travel on null geodesics, which means they only travel on "straight" trajectories, but these trajectories can themselves be bent by the presence of mass. By definition in Einstein gravity (i.e., no extra fields or modifications to the field equations), no information can travel faster than the speed of light without being acausal. Gravitational waves are a type of radiation, meaning it is the transportation of gravitational energy across spacetime. Thus, gravitational waves are predicted to travel at the speed of light in Einsteinian gravity. Read this wiki article for more details.
This was confirmed by multi-messenger observations of the neutron-star binary coalescence GW170817, where the gravitational waves emitted by the merger of the neutron stars was observed by LIGO/Virgo, and the electromagnetic counterpart of the merger was observed 1.7 seconds later by telescopes because the light can be slowed down from traveling through diffuse gas in space whereas the gravitational waves traveled at the speed of light to within 1 part in $10^{15}$. So, for all intents and purposes, gravitational waves travel at the speed of light in vacuum. In physical cosmology, speed is not simply distance/time, because it depends on your choice of coordinates and on your choice of distance measure.
answered May 4, 2022 at 16:45